Group #34034020 — subgroups

Subgroups — #34034020

Order 18688.

214 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22627061	C₂	1	2	yes	maximal
2	#22626991	C₂	1	1	yes	maximal, characteristic
3	#22626980	C₂	1	2	yes	maximal
4	#22626950	C₂	1	1	yes	maximal, characteristic
5	#22626328	C₂	1	1	yes	maximal, characteristic
6	Q₈×#10596842	—	2	2	no	—
7	#17989682	C₂²	1	1	yes	characteristic
8	#17989676	C₂²	1	2	yes	—
9	Q₈⋊#10596842	—	2	2	no	—
10	#17989671	C₂²	1	1	yes	characteristic
11	Q₁₆×#64408	C₂²	1	2	yes	—
12	#17989514	C₄	1	1	yes	characteristic
13	#17989500	C₄	1	2	yes	—
14	#17989474	C₂²	1	1	yes	characteristic
15	Q₃₂×D₇₃	C₄	1	1	yes	characteristic
16	#14556686	—	2	1	no	—
17	C₄⋊#10596842	—	2	2	no	—
18	C₄⋊#10596842	—	4	2	no	—
19	C₄×#10596842	—	4	2	no	—
20	C₄×#10596842	D₄	1	2	yes	—
21	Q₈×#64408	—	2	2	no	—
22	#14556664	C₂×C₄	1	1	yes	characteristic
23	#14556660	C₂×C₄	1	2	yes	—
24	#14556654	—	2	2	no	—
25	C₈×#64408	C₂³	1	1	yes	characteristic
26	Q₁₆×D₇₃	C₂×C₄	1	2	yes	—
27	Dic₅₈₄	C₈	1	1	yes	characteristic, complemented
28	#14556537	C₈	1	2	yes	complemented
29	C₁₆×D₇₃	C₂×C₄	1	1	yes	characteristic
30	Q₃₂×C₇₃	C₈	1	1	yes	characteristic, complemented
31	C₂×#10596842	—	2	2	no	—
32	C₂×#10596842	—	4	2	no	—
33	#11924040	—	4	1	no	—
34	#11924036	#41	1	1	yes	characteristic
35	C₄⋊#64408	—	2	2	no	—
36	C₄⋊#64408	—	4	2	no	—
37	C₄×#64408	—	4	2	no	—
38	C₄×#64408	C₂×D₄	1	1	yes	characteristic
39	#11924031	—	2	1	no	—
40	Q₈×D₇₃	—	2	2	no	—
41	Dic₂₉₂	C₂×C₈	1	2	yes	—
42	#11924018	—	2	2	no	—
43	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
44	#11924003	C₂×C₈	1	1	yes	characteristic
45	Q₁₆×C₇₃	C₂×C₈	1	2	yes	—
46	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
47	C₂×#64408	—	4	2	no	—
48	C₂×#64408	AGL(1,ℤ/8)	1	2	yes	—
49	#10596842	—	4	2	no	—
50	#10596842	—	8	2	no	—
51	Q₈×C₇₃	—	2	2	no	—
52	C₄×D₇₃	—	4	2	no	—
53	C₄×D₇₃	C₄×D₄	1	1	yes	characteristic
54	Dic₁₄₆	—	2	2	no	—
55	Dic₁₄₆	—	4	2	no	—
56	#10596833	—	2	1	no	—
57	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
58	#10596831	#131	1	1	yes	characteristic
59	C₂×D₇₃	#442	1	1	yes	characteristic
60	#64408	—	2	2	no	—
61	#64408	—	8	2	no	—
62	C₄×C₇₃	—	4	2	no	—
63	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
64	Dic₇₃	—	4	2	no	—
65	Dic₇₃	#444	1	1	yes	characteristic
66	#12066	—	73	1	no	maximal, Sylow
67	C₂×C₇₃	#1580	1	1	yes	characteristic
68	D₇₃	#2180	1	2	yes	—
69	#2176	—	73	1	no	—
70	#1579	—	73	2	no	—
71	#1368	—	73	1	no	—
72	#1334	—	73	2	no	—
73	#1313	—	73	1	no	—
74	C₇₃	#12066	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₃₂	—	73	1	no	—
76	C₈×Q₈	—	146	2	no	—
77	C₄×Q₁₆	—	73	2	no	—
78	#367	—	73	1	no	—
79	#358	—	73	2	no	—
80	#346	—	73	1	no	—
81	#346	—	73	1	no	—
82	#334	—	73	2	no	—
83	Q₈⋊C₈	—	146	2	no	—
84	#322	—	73	1	no	—
85	C₂×Q₁₆	—	73	2	no	—
86	C₄×Q₈	—	146	2	no	—
87	Q₃₂	—	73	2	no	—
88	Q₃₂	—	73	1	no	—
89	Q₃₂	#10596842	1	1	yes	characteristic, complemented
90	C₂×C₁₆	—	73	1	no	—
91	C₂×C₁₆	—	73	1	no	—
92	C₂×C₁₆	—	146	1	no	—
93	#107	—	73	2	no	—
94	C₄⋊C₈	—	292	2	no	—
95	C₄⋊C₈	—	146	2	no	—
96	#103	—	146	2	no	—
97	C₄×C₈	—	292	2	no	—
98	C₄×C₈	—	73	1	no	—
99	C₄×C₈	—	73	2	no	—
100	C₂×Q₈	—	146	2	no	—
101	Q₁₆	—	73	2	no	—
102	Q₁₆	—	146	2	no	—
103	Q₁₆	C₂×#10596842	1	2	yes	—
104	C₂×C₈	—	292	2	no	—
105	C₂×C₈	—	73	1	no	—
106	C₂×C₈	—	73	1	no	—
107	C₂×C₈	—	146	2	no	—
108	C₄⋊C₄	—	292	2	no	—
109	C₄⋊C₄	—	146	2	no	—
110	C₄²	—	292	2	no	—
111	C₄²	—	73	1	no	—
112	C₁₆	—	292	1	no	—
113	C₁₆	—	73	1	no	—
114	C₁₆	—	146	1	no	—
115	C₁₆	C₂×#10596842	1	1	yes	characteristic
116	Q₈	—	2	2	no	—
117	Q₈	—	292	2	no	—
118	Q₈	—	146	2	no	—
119	C₂×C₄	—	292	2	no	—
120	C₂×C₄	—	292	2	no	—
121	C₂×C₄	—	73	1	no	—
122	C₂×C₄	—	73	2	no	—
123	C₈	—	292	2	no	—
124	C₈	—	584	2	no	—
125	C₈	—	73	1	no	—
126	C₈	—	146	1	no	—
127	C₈	C₂²×#10596842	1	1	yes	characteristic
128	C₂²	—	73	1	no	—
129	C₄	—	4	2	no	—
130	C₄	—	292	2	no	—
131	C₄	—	584	2	no	—
132	C₄	—	73	1	no	—
133	C₄	—	146	2	no	—
134	C₄	D₄×#10596842	1	1	yes	characteristic
135	C₂	—	73	2	no	—
136	C₂	#22627062	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627061

C₂

yes

maximal

#22626991

C₂

yes

maximal, characteristic

#22626980

C₂

yes

maximal

#22626950

C₂

yes

maximal, characteristic

#22626328

C₂

yes

maximal, characteristic

Q₈×#10596842

—

#17989682

C₂²

yes

characteristic

#17989676

C₂²

yes

—

Q₈⋊#10596842

—

#17989671

C₂²

yes

characteristic

Q₁₆×#64408

C₂²

yes

—

#17989514

C₄

yes

characteristic

#17989500

C₄

yes

—

#17989474

C₂²

yes

characteristic

Q₃₂×D₇₃

C₄

yes

—

—

—

—

D₄

yes

—

Q₈×#64408

—

#14556664

C₂×C₄

yes

characteristic

#14556660

C₂×C₄

yes

—

#14556654

—

C₈×#64408

C₂³

yes

characteristic

Q₁₆×D₇₃

C₂×C₄

yes

—

Dic₅₈₄

C₈

yes

characteristic, complemented

#14556537

C₈

yes

complemented

C₁₆×D₇₃

C₂×C₄

yes

characteristic

Q₃₂×C₇₃

C₈

yes

characteristic, complemented

—

—

—

yes

—

—

—

C₂×D₄

yes

characteristic

#11924031

—

Q₈×D₇₃

—

Dic₂₉₂

C₂×C₈

yes

—

#11924018

—

C₈×D₇₃

C₂²×C₄

yes

characteristic

#11924003

C₂×C₈

yes

characteristic

Q₁₆×C₇₃

C₂×C₈

yes

—

C₁₆×C₇₃

C₂×C₈

yes

—

yes

—

#10596842

—

#10596842

—

Q₈×C₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₄×D₄

yes

characteristic

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

C₈×C₇₃

C₂²×C₈

yes

yes

C₂×D₇₃

yes

—

—

C₄×C₇₃

—

C₄×C₇₃

C₈×D₄

yes

characteristic

Dic₇₃

—

Dic₇₃

yes

—

C₂×C₇₃

yes

D₇₃

yes

—

—

—

—

—

—

C₇₃

#12066

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₈×Q₈

—

146

—

C₄×Q₁₆

—

—

—

—

—

—

Q₈⋊C₈

—

146

—

#322

—

C₂×Q₁₆

—

C₄×Q₈

—

146

—

Q₃₂

—

Q₃₂

—

Q₃₂

#10596842

yes

characteristic, complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

#107

—

C₄⋊C₈

—

292

—

C₄⋊C₈

—

146

—

#103

—

146

—

C₄×C₈

—

292

—

C₄×C₈

—

C₄×C₈

—

100

C₂×Q₈

—

146

—

101

Q₁₆

—

102

Q₁₆

—

146

—

103

Q₁₆

C₂×#10596842

yes

—

104

C₂×C₈

—

292

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

146

—

108

C₄⋊C₄

—

292

—

109

C₄⋊C₄

—

146

—

110

C₄²

—

292

—

111

C₄²

—

112

C₁₆

—

292

—

113

C₁₆

—

114

C₁₆

—

146

—

115

C₁₆

C₂×#10596842

yes

characteristic

116

Q₈

—

117

Q₈

—

292

—

118

Q₈

—

146

—

119

C₂×C₄

—

292

—

120

C₂×C₄

—

292

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₈

—

292

—

124

C₈

—

584

—

125

C₈

—

126

C₈

—

146

—

127

C₈

C₂²×#10596842

yes

characteristic

128

C₂²

—

129

C₄

—

130

C₄

—

292

—

131

C₄

—

584

—

132

C₄

—

133

C₄

—

146

—

134

C₄

D₄×#10596842

yes

characteristic

135

C₂

—

136

C₂

#22627062

yes

characteristic

	subgroup	G/H
center	C₂	#22627062
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₈	C₂²×#10596842
Fitting	Q₃₂×C₇₃	C₈
socle	C₂×C₇₃	#1580
prime radical	C₂×D₇₃	#442
cosocle	C₈×#64408	C₂³
hypercenter	Q₃₂	#10596842
hypocenter	C₇₃	#12066
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

subgroup

G/H

center

C₂

#22627062

commutator

C₈×C₇₃

C₂²×C₈

Frattini

C₈

C₂²×#10596842

Fitting

Q₃₂×C₇₃

C₈

C₂×C₇₃

C₂×D₇₃

C₂³

Q₃₂

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

N	H ≅ G/N	type
27. Dic₅₈₄	C₈	semidirect
28. #14556537	C₈	semidirect
30. Q₃₂×C₇₃	C₈	semidirect
74. C₇₃	#12066	semidirect
89. Q₃₂	#10596842	semidirect

H ≅ G/N

type

27. Dic₅₈₄

C₈

semidirect

28. #14556537

C₈

semidirect

30. Q₃₂×C₇₃

C₈

semidirect

74. C₇₃

#12066

semidirect

89. Q₃₂

#10596842

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22627061	2
#22626991	2
#22626980	2
#22626950	2
#22626328	2
#12066	73

subgroup

index

Subgroups — #34034020

Conjugacy classes of subgroups

Complements

Maximal subgroups